Mathematics for the interested outsider

Open thread

Tomorrow I’m going to be gone pretty much all day. Dartmouth is a nice, straight, three-hour shot up I-91, so I’ll head out in the morning, spend the afternoon, and return at night. No sense paying for lodging or anything if I don’t have to.

Anyhow, since I won’t be able to post, I thought I’d leave this thread open. What’s on your mind? Anything you’d like to see treated here? Just who the hell are you people, anyway?

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Who am I? I’m an IT guy rediscovering my mathematical roots (more details at my blog…) I am also an amateur astronomer and currently have a 17.5″ dobsonian.

I’ve been following your Group Theory posts – your post on Cosets really made a ‘light bulb’ come on for me. Especially the description – ‘A subgroup is sort of like a slice through a group. The cosets of a subgroup are like slices “parallel” to the group’.

I’m going to work through the ‘The First Isomorphism Theorem’ in conjunction with the relevant section of Artin’s Algebra when I get time. It looks like an interesting and important part of the theory.

I’m a low-dimensional topology grad student at UCSB, but this year I’m teaching at Westmont college. I believe that I actually sat near you at some of the talks at the NE sectional meeting of the AMS last year at Dartmouth, though I don’t think we ever actually met. I’m enjoying your blog so far…

I’m a final year biochemistry student at the moment, hopefully going on to do a PhD in something on the chemistry side of my subject. I read a bit of your blog a while back, I only read the bits I’m interested in (which is mostly category theory), but got confused around “MacLane’s Coherence Theorem”.

I came back because I learned read an enriched category is (in “arrows structures and functors: The Categorical Imperative”, although they called it a V-Category). Before learning that I found hom sets, the Yoneda lemma etc. rather boring (not just when I read them on your blog, also in the two books on category theory I’ve been reading), so decided to just stick with the best book. Now hom sets are suddenly interesting.

But I have a question. Have you covered anywhere about how fields fit with category theory (I can’t find any in your archives). I see they are rings, so you can talk about them in terms of ring algebras over the category of sets and functions. To talk about division you have to go to the category of sets and partial functions though – it just doesn’t look nice. But I can see that I’m probably missing something really cool. If you haven’t then that’s my request, although my opinion shouldn’t be worth tuppence ’round here, as I could well stop reading before you start (finals!) or not understand the discussion because I haven’t read most of the last few years of archives.

I have to admit, I really haven’t done much with fields. Really the only thing I know how to do with much depth in field theory is basic Galois theory, and I had other things I wanted to do first. I’ll probably hit Galois theory at some point, sure, but not yet.

One thing I can note now, since you’re interested in category theory, is that there’s something called a “Galois correspondence”, which basically amounts to a pair of direction-reversing maps of posets with a certain other property. But we know that posets are categories, and a direction-reversing map of posets is a contravariant functor. It turns out the extra condition is basically that the two functors are adjoints, and then certain other nice things happen. Once you have this viewpoint set up, a lot of basic Galois theory amounts to setting up some definitions, and the interesting conclusions just fall out of the category theory.

As for division, though.. division is always weird. There’s even a pretty heavy paper out there just on dividing by three!

About this weblog

This is mainly an expository blath, with occasional high-level excursions, humorous observations, rants, and musings. The main-line exposition should be accessible to the “Generally Interested Lay Audience”, as long as you trace the links back towards the basics. Check the sidebar for specific topics (under “Categories”).

I’m in the process of tweaking some aspects of the site to make it easier to refer back to older topics, so try to make the best of it for now.